On parametric hypotheses testing with nonparametric tests

Yu. N. Tyurin

Teoriâ veroâtnostej i ee primeneniâ, Tome 15 (1970) no. 4, pp. 745-749

Résumé

Let $\xi_1,\dots,\xi_n$ be a sample with a theoretical distribution function s$F(t,\theta)$. Here $\theta$ is an unknown $r$-dimensional parameter. We consider the “regular” case when its maximum likelihood estimator $\theta^*$ has usual asymptotical properties. We denote the empirical distribution function by $F_n(t)$. In this paper, limiting properties of $\sqrt n[F_n(t)-F(t,\theta^*)]$ are discussed. It is proved that $\lim\sqrt n[F_n(t)-F(t,\theta^*)]$ is a conditioned Gaussian process. After a natural change of the time variable $s=F(t,\theta^*)$ we obtain a conditioned Wiener process $v(s)$ on $[0,1]$ satisfying $r$ linear conditions $$ \int_0^1m_i(s,\theta)\,dv(s)=0,\quad i=1,\dots,r, $$ and $v(1)=0$. If $\theta$ is the location-scale parameter the conditions are free of $\theta$. A linear transformation $v\to\tilde v$ is constructed, where the Wiener process $\tilde v(s)$ satisfies $r+1$ conditions: $$ \tilde v(0)=\tilde v(t_1)=\dots=\tilde v(t_r)=\tilde v(1)=0. $$ Quantities $0<t_1<\dots<t_r<1$ can be chosen arbitrarily. Now it is possible to use for the process $\tilde v$ such well-known goodness-of-fit tests as Kolmogorov's or Cramer–von Mises' ones.